For instance, real numbers are dense, between every two real numbers there is another. E.g., halfway between 0.003 and 0.004 there is 0.0035. So there is a real number at every point along the number line.

But integers are discrete; between ten and eleven there are no integers.

The ancient Greek philosophers debated whether matter was continuous or not; some suggested that given a lump of iron, you could continue splitting it in half indefinitely and just get smaller and smaller lumps of iron - less stuff, but still the same kind of stuff. But others like Democritus suggested that there was an indivisible unit, "atomos" in Greek. We now know that matter is discrete; once you get down to one iron atom it's a lot harder to cut in half, and if you do you will break it; you'll get electrons, protons and neutrons, none of which are iron.

In general, if something is "discrete", it can be broken into distinct parts or steps.

The word "discrete" is from Latin, and is related to "concrete" and "accretion", with the negative prefix "dis". So the Latin means something like "not grown together".

Discrete movement. See Concrete movement of the voice, under Concrete, a. -- Discrete proportion, proportion where the ratio of the means is different from that of either couplet; as, 3:6::8:16, 3 bearing the same proportion to 6 as 8 does to 16. But 3 is not to 6 as 6 to 8. It is thus opposed to continued or continual proportion; as, 3:6::12:24. -- Discrete quantity, that which must be divided into units, as number, and is opposed to continued quantity, as duration, or extension.